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I know of examples of periodic continuous functions whose Fourier series diverge on sets of measure zero. Is it possible for a Fourier series of a periodic continuous function to converge to something other than the function value, at some point? If so, can someone provide an example or reference?

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  1. If a series converges to a limit $L$, its Cesaro averages converge to the same limit $L$.

  2. The Cesaro averages of a Fourier series are given by convolution with the Fejer kernel, and it is known that convolving an element of $C({\bf T})$ with the Fejer kernel always gives a sequence of trigonometric polynomials converging uniformly to the given function.

So I think the answer to your question is negative: if $f$ is continuous, and if the partial sums of $\sum_{n\in{\bf Z}} \hat{f}(n)e^{2\pi int}$ converge to some $L$, then $L=f(t)$.

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    $\begingroup$ Yes, and the result itself is due to Fejér: en.wikipedia.org/wiki/Fej%C3%A9r%27s_theorem $\endgroup$ – GH from MO Apr 14 '17 at 14:28
  • $\begingroup$ Thanks for the answer. What about the following generalization of this question for discontinuous functions: if the Fourier series converges to L at some point, does L have to be within the oscillation interval of the function at that point? $\endgroup$ – Zbigniew Fiedorowicz Apr 14 '17 at 14:33
  • $\begingroup$ And, in particular, if there is a jump discontinuity at t, does L have to be the center point of the oscillation interval? $\endgroup$ – Zbigniew Fiedorowicz Apr 14 '17 at 14:45
  • $\begingroup$ @GHfromMO Yes, I should have stated that point number 2 is Fejer's theorem. Point number 1 would have been known to him and well before him $\endgroup$ – Yemon Choi Apr 14 '17 at 14:48
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    $\begingroup$ @Yemon Choi I guess I am thinking of a function which is continuous except for a finite set of discontinuities within a periodicity interval. $\endgroup$ – Zbigniew Fiedorowicz Apr 14 '17 at 14:54

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